IOQM Composite Numbers Question, Download Practice PDF

In the Indian Olympiad Qualifier in Mathematics (IOQM), Number Theory plays a huge role in the exam. Many students focus only on primes. But they forget that composite numbers are equally important. In fact, many IOQM composite numbers questions test your understanding of factors, divisibility, floor function, inequalities, and integer properties.

Mastering these problem types helps students improve their conceptual understanding and problem-solving skills for IOQM. Therefore, students must solve IOQM composite numbers questions. Find here IOQM composite number problems with solutions, along with tips to use them for the upcoming exam/.

Also Read: 

• IOQM Previous Year Question Papers
• IOQM Syllabus

Introduction of IOQM Composite Numbers

Number Theory is a fundamental branch of mathematics. It explores properties of integers, including prime and composite numbers, divisibility rules, and special functions. While IOQM composite numbers questions are common, there is another crucial segment: problems using the Greatest Integer Function (also known as the floor function) and the Fractional Part Function. 

Solving IOQM Number Theory questions helps students practice and understand common question patterns related to these functions. It offers detailed solutions to complex problems, building a strong foundation in advanced number theory concepts.

Important Questions for IOQM Composite Numbers

Here is a collection of important questions related to Number Theory, specifically focusing on the Greatest Integer Function and Fractional Part Function for IOQM exams. 

1. What is the greatest integer less than or equal to

2. The symbolism [x] denotes the largest integer not exceeding x. For example, [3] = 3 and [9/2] = 4.

Compute:

[√1] + [√2] + [√3] + ... + [√16]

3. If x and y satisfy the equations

y = 2[x] + 9

and

y = 3[x + 2]

simultaneously, where [.] denotes the greatest integer function, then find the value of [x + y].

4. The number of solution(s) of

{x} + {x²} + {x³} = 3

is ______

(where {. } denotes the fractional part function).

5. Find the number of integral value(s) of x satisfying the inequality

2|x + 2| – |x + 5| ≤ 4

6. Find the number of integral values of x satisfying

[x² – 2]²  – 9 [x² – 2] + 14 = 0

where [.] denotes the greatest integer function.

7. For each real number x, let [x] denote the greatest integer that does not exceed x. Let Q be the number of values of positive integers n such that it is true that n < 1000 and [log₂ n] is a positive even integer.

Find Q – 300.

8. Let

f(x) = sin([π²]x) + cos([–π²]x)

where [.] denotes the greatest integer function.

Find f(π/2).

9. Let z = [y] and y = [x] – x, where [.] denotes the greatest integer function. If x is not an integer but positive, then what is the absolute value of |z|?

10. Calculate the value of [p] if

f(x) = [1.2]p + [–2.23]p + 3[1.2p] = 0

where [.] denotes the greatest integer function.

IOQM Composite Numbers Question with Solution PDF

A downloadable PDF containing these important questions and their solutions for IOQM Number Theory: Greatest Integer Function is available. Students can download this resource to practice offline and strengthen their preparation effectively.

Download IOQM Composite Numbers Question with Solution PDF

How to use Important Solutions for IOQM Number Theory for Preparation?

Prepare for your IOQM exam effectively using these important solutions with the following steps:

• Review Concepts Thoroughly – Before attempting problems, ensure a solid grasp of Number Theory principles, especially properties of the Greatest Integer and Fractional Part Functions.

• Practice Questions Actively – Solve each problem without looking at the solution first. This tests your understanding and problem-solving skills independently.

• Analyze Solutions Carefully – After solving, compare your approach with the provided solutions. Understand every step and identify any shortcuts or alternative methods.

• Identify Weak Areas – Note down question types or concepts where you struggle. Dedicate extra study time to these areas to strengthen your foundation.

• Timed Practice – Incorporate timed practice sessions using these questions to improve speed and accuracy, simulating actual exam conditions.

• Revise Regularly – Periodically revisit these questions to reinforce learning and ensure long-term retention of problem-solving techniques.